Algebra PYQ — Page 40

For $z\in \mathbb{C}$ if the minimum value of $(|z-3\sqrt{2}|+|z-p\sqrt{2}i|)$ is $5\sqrt{2}$, then a value of $p$ is _______.

If the sum of the co-efficients of all the positive even powers of $x$ in the binomial expansion of ${(2{x}^{3}+\frac{3}{x})}^{10}$ is ${5}^{10}-\beta \cdot {3}^{9}$, then $\beta$ is equal to _____.

The sum of all the elements of the set ${\alpha \in {1,2,\ldots ..100}:HCF(\alpha ,24)=1}$ is

If $x=\sum _{n=0}^{\infty }{a}^{n},y=\sum _{n=0}^{\infty }{b}^{n},z=\sum _{n=0}^{\infty }{c}^{n}$, where $a,b,c$ are in A.P. and $|a|<1,|b|<1,|c|<1$, $abc\neq 0$, then

If the constant term in the expansion of ${(3{x}^{3}-2{x}^{2}+\frac{5}{{x}^{5}})}^{10}$ is ${2}^{k}.l$, where $l$ is an odd integer, then the value of $k$ is equal to

The area of the polygon, whose vertices are the non-real roots of the equation $\bar{z}=i{z}^{2}$ is

Suppose ${a}_{1},{a}_{2},\ldots ,{a}_{n},\ldots$ be an arithmetic progression of natural numbers. If the ratio of the sum of the first five terms to the sum of first nine terms of the progression is $5:17$ and $110<{a}_{15}<120$, then the sum of the first ten terms of the progression is equal to

If ${a}_{1}(>0),{a}_{2},{a}_{3},{a}_{4},{a}_{5}$ are in a G.P. , ${a}_{2}+{a}_{4}=2{a}_{3}+1$ and $3{a}_{2}+{a}_{3}=2{a}_{4}$, then ${a}_{2}+{a}_{4}+2{a}_{5}$ is equal to _____.

Let the matrix $A=[\begin{matrix}0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1\end{matrix}]$ and the matrix ${B}_{0}={A}^{49}+2{A}^{98}$. If ${B}_{n}=\mathrm{Adj}({B}_{n-1})$ for all $n\geq 1$, then $det({B}_{4})$ is equal to

The number of real values of $\lambda$, such that the system of linear equations $2x-3y+5z=9$ $x+3y-z=-18$ $3x-y+({\lambda }^{2}-|\lambda |)z=16$ has no solutions, is

Let $A$ be a matrix of order $3\times 3$ and $\mathrm{det}(A)=2$. Then $\mathrm{det}(det(A)adj(5adj({A}^{3}))$ is equal to _____.

Let $A=[\begin{matrix}2 & -1 & -1 \\ 1 & 0 & -1 \\ 1 & -1 & 0\end{matrix}]$ and $B=A-I$. If $\omega =\frac{\sqrt{3}i-1}{2}$, then the number of elements in the set ${n\in {1,2,\ldots ,100}:{A}^{n}+{(\omega B)}^{n}=A+B}$ is equal to _____ .

Let $x,y>0$. If ${x}^{3}{y}^{2}={2}^{15}$, then the least value of $3x+2y$ is

Let $S={x\in [-6,3]-{-2,2}:\frac{|x+3|-1}{|x|-2}\geq 0}$ and $T={x\in Z:{x}^{2}-7|x|+9\leq 0}$. Then the number of elements in $S\cap T$ is

Let $f(x)$ be a quadratic polynomial with leading coefficient $1$ such that $f(0)=p,p\neq 0$, and $f(1)=\frac{1}{3}$. If the equations $f(x)=0$ and $fofofof(x)=0$ have a common real root, then $f(-3)$ is equal to ______.

Let the sum of an infinite $G.P.$, whose first term is $a$ and the common ratio is $r$, be $5$. Let the sum of its first five terms be $\frac{98}{25}$. Then the sum of the first $21$ terms of an $\mathrm{AP}$, whose first term is $10ar,{n}^{\mathrm{th}}$ term is ${a}_{n}$ and the common difference is $10{ar}^{2}$, is equal to

Let $f:R\rightarrow R$ be a continuous function such that $f(3x)-f(x)=x$. If $f(8)=7$, then $f(14)$ is equal to:

Let $S={1,2,3,4}$. Then the number of elements in the set {$f:S\times S\rightarrow S:f$ is onto and $f(a,b)=f(b,a)$ $\geq a\forall (a,b)\in S\times S$} is

Let $S$ be the set containing all $3\times 3$ matrices with entries from ${-1,0,1}$. The total number of matrices $A\in S$ such that the sum of all the diagonal elements of ${A}^{T}A$ is $6$ is ______.

There are ten boys ${B}_{1},{B}_{2},\ldots .,{B}_{10}$ and five girls ${G}_{1},{G}_{2},\ldots .{G}_{5}$ in a class. Then the number of ways of forming a group consisting of three boys and three girls, if both ${B}_{1}$ and ${B}_{2}$ together should not be the members of a group, is _____.

Let $f(x)$ be a quadratic polynomial such that $f(-2)$ $+f(3)=0$. If one of the roots of $f(x)=0$ is $-1$, then the sum of the roots of $f(x)=0$ is equal to

Let ${b}_{1}{b}_{2}{b}_{3}{b}_{4}$ be a $4$-element permutation with ${b}_{i}\in$ ${1,2,3,\ldots \ldots \ldots ,100}$ for $1\leq i\leq 4$ and ${b}_{i}\neq {b}_{j}$ for $i\neq j$, such that either ${b}_{1},{b}_{2},{b}_{3}$ are consecutive integers or ${b}_{2},{b}_{3},{b}_{4}$ are consecutive integers. Then the number of such permutations ${b}_{1}{b}_{2}{b}_{3}{b}_{4}$ is equal to ______.

Let $S={(\begin{matrix}-1 & a \\ 0 & b\end{matrix});a,b\in {1,2,3,\ldots 100}}$ and let ${T}_{n}={A\in S:{A}^{n(n+1)}=I}$. Then the number of elements in $\cap _{n=1}^{100}{T}_{n}$ is _____.

The domain of the function $f(x)={\mathrm{sin}}^{-1}(\frac{{x}^{2}-3x+2}{{x}^{2}+2x+7})$ is